During discussion on another question, the question of singularties in modern physical theories arose. The big-bang is an obvious singularity in modern conceptions of the cosmos. Is it a goal of modern theories to find a way to remove the big bang singularity from theory, and how is that accomplished in theories being proposed today?

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I think it is not quite true that modern physics tries to "avoid Big Bang singularities". Among many other things, modern physics tries to determine what is happening in the vicinity of the region that the Big Bang cosmology views as a singularity.

By saying that the Big Bang singularity is completely avoided, you are already presupposing an answer. This possible answer may be a "working hypothesis" but it may also be incorrect. So it is not a "goal of modern physics".

Classical general relativity breaks down - and becomes unpredictive - in the presence of singularities. But that doesn't mean that the full theory that also knows everything about the short-distance physics has to avoid the singularities completely.

In string theory, the Big Bang singularity hasn't been understood yet - at least to the satisfaction of most string theorists. But many other singularities have been fully understood and it is not true that all of them disappeared. Some of them didn't disappear but the physics around them began well-defined, anyway.

In particular, time-like singularities - such as orbifold singularities; orientifold singularities; and conifold points - have been pretty much fully understood. There are new degrees of freedom and new phenomena that take place in their vicinity but in some sense, physics understands them as well as it understands the smooth space today.

From some viewpoint - e.g. according to some "probes" (objects whose reactions we use to evaluate what's happening in the region) - the singularity may get regulated, replaced by a regular manifold with some typical length scales. Other probes see the singularity replaced by a non-commutative geometry that is also regulated. But some singularities according to some probes still look singular; the best geometrical description is still a geometry that has a full-fledged singularity at the original point. However, it is not necessarily a problem. Despite the singularity, physics at some (not quite) manifolds such as the conifold may be shown to be completely equivalent to physics at a completely smooth manifold (e.g. by mirror symmetry).

This equivalence shows that physics at singular manifolds may be well-defined and predictive.

The space-like singularities - like the singularity inside the Schwarzschild black hole and the Big Bang singularity - remain confusing. Some people even think that all questions involving the interior of a black hole or the very beginning of the Universe are inevitably ill-defined, at least to some extent, so there will never be a set of sharp observables that can be exactly calculated and discussed. I am personally not sure whether it is the case: it could be.

But of course, the working hypotheses what's happening near the Big Bang - probably before inflation - also include some models with bounces; cyclic Universes; non-commutative geometry; a beginning of the Universe from "zero" that looks totally smooth after the Wick rotation to the Euclidean space (the Hartle-Hawking state); but also an abrupt tunneling into another Universe, and many other things. The collection of possibilities is pretty rich and some of them are more motivated than others. Of course, physics still doesn't know for sure which of the tools are truly relevant for the birth of the Universe.